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Metamaterials with Interacting Metaatoms and Spatial Dispersion Metamaterials with interacting Metaatoms A. Chipouline, S. Sugavanam, J. Petschulat, T. Pertsch Institute of Applied Physics, Friedrich-Schiller-Universität Jena, Max Wien Platz 1, 07743, Jena, Germany Abstract An analytical model for homogenization of Metamaterials (MM) with interacting Metaatoms (MA) is developed based on multipole approach for bulk media. The interaction is assumed to be near field type. i.e. no retardation in lateral direction between adjacent MAs is assumed. The interaction takes place between the adjacent MAs only in lateral (perpendicular to the propagation) direction; the considered MM is supposed to consist of non-interacting identical layers with periodically spaced MAs. It is shown, that the interaction between MAs leads to significant changes of dispersion characteristics, in particular, appearance of the spatial dispersion is emphasized. The results indicate increase of the available real part of the k vector values and increase of imaginary part by the approximately same percentage, which leads to the final conclusion that the effect of coupling does not provide extra opportunities for the resolution enhancement. 1. Introduction Volume or statistical averaging of the microscopic Maxwell equations (MEs), i.e. transition from microscopic MEs to their macroscopic counterparts, is one of the main steps in electrodynamics of materials. In spite of the fundamental importance of the averaging procedure, it is quite rarely properly discussed in university courses and respective books; up to now there is no established consensus about how the averaging procedure has to be performed. In [1] it has been show that there are some basic principles for the averaging procedure (irrespective to what type of material is studied) which have to be satisfied in order to be accepted as a credible one. In this paper show how the multipole approach in homogenization of the MM [2, 3] can be extended on the case of regularly placed interacting MA in form of the double wire structure [4]. The multipole expansion and effective medium constants elaboration, developed in [5], result in constructive expressions for multipoles as functions of microscopic charge dynamics. This solves the problem of homogenization, provided the microscopic charge dynamics is expressed through the averaged (macroscopic) electric and magnetic fields. Nevertheless, during the elaboration of this approach, a lot of assumptions have to be accepted, and many of them can be rather poor justified for the typical MM configurations. For example, assumptions about significantly large distance between atoms in compare with the atomic sizes is undoubtedly good satisfied in case of solid states, but for MM, where MA are just maximum five times smaller (for typical structures) then the distance between them, this approximation is rather questionable. Nevertheless, it is believed that the approach used in [3] is satisfactory good (at least as a first approximation); moreover, the fitting parameters used in the model absorbs to some extend the mentioned above deviation between rigorous representation [5] and one elaborated in [3]. The approach presented in [3] has a lot of potential for further development due to the fact that the expressions for microscopic dynamics of charges can be elaborated, taking into account interaction between MA (which is a subject of this paper), and/or MA of more complex structure, consisting of not only plasmonic nano resonators, but, for example, the plasmonic nano resonators coupled with quantum systems, like Carbon Nanotubes [6] or Quantum Dots [7]. In this paper we consider a metamaterial consisting of the identical layers with regularly spaced MA in each; one layer, which the MM consists of (the layers repeat themselves in y direction), is presented in Fig. 1. Fig. 1: Artificial metaatoms (plasmonic nano resonators) embedded in a dielectric matrix form a MM (only one layer is presented). Polarization of the electric and magnetic fields, and direction of the wave vector are shown. The interaction between MAs takes place in z direction; the possible interaction in y direction (wave propagation direction) is not taken into account. The interaction between MAs is assumed to be negligible in longitudinal direction (perpendicular to the layer surfaces) or, in other words, the layers are assumed to be good separated from each other. The effect of interaction in lateral direction (parallel to the layer surface) and its influence on the dispersion relation of the plane waves propagating in the MM consisting of the interacting MAs, is a subject of the presented paper. When there is a coupling between MA, the response of the medium is no longer truly local. As a result, depending upon the configuration of the MA, the medium responds differently to electromagnetic waves propagating in different directions. This phenomenon is called spatial dispersion (akin to the phenomenon of temporal dispersion – where the response of a medium at a given time depends on the history of its excitation). The direction dependence of the medium response is not just unique for the spatial dispersion – in case of anisotropic media this effect appears as well. The qualitative markers for differentiation of these two effects has been considered in details in [9] and [10] (see also references therein). First, the effect of anisotropy which leads to different wave vectors for different propagation directions and which is typical for crystals, has to be taken into account. It was shown that the effect of anisotropy leads to two possible families of isofrequency contours, namely ellipsoids and hyperboloids, ε ε depending on the sign of the ratios zz and yy (here the consideration is carried out in the main axis εxx εxx of the intrinsic coordinate system of the crystal) [11], [12], [13]; the effective parameters εxx , ε yy , and εzz themself do not depend on the wave vector (spatial dispersion is negligible). In media with spatial dispersion the effective parameters depend on the wave vector considerably, therefore making the shape of the isofrequency contours arbitrary. Comparison of isofrequencies of MM obtained numerically with ellipsoids and hyperboloids makes it possible to identify frequency intervals, where the effective parameters do not depend on the wave vector. One of the ways to describe spatial dispersion is to use a model of a chain of the coupled harmonic oscillators. Such a model is adequate as a first approximation for the interactions between plasmonic oscillations in the metallic nano-resonators. Eigen modes of the response are obtained as wave solutions giving the oscillations of the plasmonic charges (see Fig. 2). Fig. 2: Spatial dispersion viewed as a consequence of a coupling effect in a chain of dipoles. The problem is equivalent to the study of transverse oscillation dynamics in a chain of the coupled harmonic oscillators. Similar approach has been used to study effect of interacting of the MAs on the properties of the MM in [14-17] for microwave frequency region. To better understand the problem, first the case of a one dimensional chain of coupled harmonic oscillators (coupled dipoles) is studied, after which the problem is extended to the case of coupled MA in form of cut-wires (coupled quadrupoles). The results of the analysis for both ensembles are presented side by side to enable a comparative study of the dispersion characteristics. The knowledge of the dispersion relation is very important; actually, it is the only, what is required to analyze the wave propagation in a media (boundary condition problems are not included in the discussion here). For example, it is known that in order to provide a better resolution the media has to allow propagation of the lateral components of the wave vector k with as much as possible values (as higher as possible kz components in Fig. 3 for the same wavelength), which could be achieved, for example, in a hyperbolic dispersion media [18]. The performed below analysis provides a tool to analyze whether the coupling between MAs could enhance the available spatial spectrum of the propagating waves and hence increase resolution of the optical systems, which use the respective MM. The paper consists of five parts. After introduction (part 1), the dispersion relations for eigen waves in chains of coupled dipoles and quadrupoles will be summarized in part 2. The dispersion relations in form of transcendental equations for the electro-magnetic plane waves propagating in the media, consisting of the coupled dipoles and quadrupoles will be elaborated in part 3. In part 4 the respective solutions will be provided and the results will be summarized; short conclusion is given in part 5. 2. Dispersion relations for material eigen waves 2.1 Periodic chain of coupled dipoles A chain of periodically positioned dipoles (oriented with the long axis along the x direction) is considered (see Fig. 3). For clarity, only one row is shown in the figure; it is assumed that rows of dipoles are placed along the y direction. The treatment of the problem remains the same, and coupling between neighboring rows are neglected. The arrangement of the dipoles is along the z direction; the period of spacing is taken to be z0 . The effect of coupling with adjacent oscillators is introduced via a coupling constant σ that is a function of the distance of separation between the oscillators. Well known solution in form of the transverse spatial modes which can be sustained in such a medium under the above mentioned conditions can be
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